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Completeness theorem for logic with imprecise and conditional probabilities. (English) Zbl 1144.03019
Building upon earlier work of the authors, this interesting paper offers an axiomatic treatment for precise as well as imprecise absolute and conditional probabilities within the realm of a propositional logic. The authors assume to have given a recursive nonarchimedean field \(S\) which contains all the rationals from the real unit interval, and also an infinitesimal. The language of the axiomatic system is enriched with different unary probability operators having each one of the elements from this field \(S\) as a parameter. Additionally the system has some infinitary inference rules.
For this system an adequate algebraic semantics is provided, and the completeness theorem proven. The semantic structures are Kripke frames with total accessibility relation which provide each possible world with a probability space with \(S\)-valued probability measures.

MSC:
03B48 Probability and inductive logic
03B70 Logic in computer science
03B45 Modal logic (including the logic of norms)
68T37 Reasoning under uncertainty in the context of artificial intelligence
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